In many sensing applications, such as radar pulse compression, sonar, ultrasonic non-destructive evaluation for structural integrity, biomedical imaging, and seismic estimation, it is desirable to estimate the impulse response of an unknown system by driving the system with a known signal having a finite temporal extent. The process of separating the known signal from the received output of the system in order to estimate the unknown impulse response is known as predictive deconvolution.
Predictive deconvolution provides a means to obtain the high spatial resolution of a short, high bandwidth pulse without the need for very high peak transmit power, which may not be feasible. This is accomplished by transmitting a longer pulse that is phase or frequency modulated to generate a wideband waveform. The transmission of the wideband waveform into the unknown system results in a received return signal at the sensor that is the convolution of the waveform and the system impulse response, which possesses large coefficient values at sample delays corresponding to the round-trip travel time of the transmitted waveform from the sensor to a significant reflecting object (also called a scatterer) and back to the sensor. The purpose of predictive deconvolution is to accurately estimate the unknown system impulse response from the received return signal based upon the known transmitted waveform.
A well-known approach to predictive deconvolution, used extensively in radar and biomedical imaging applications, is known as matched filtering, e.g. as described in M. I. Skolnik, Introduction to Radar Systems, McGraw-Hill, New York, 1980, pp. 420-434; T. X. Misaridis, K. Gammelmark, C. H. Jorgensen, N. Lindberg, A. H. Thomsen, M. H. Pedersen, and J. A. Jensen, “Potential of coded excitation in medical ultrasound imaging,” Ultrasonics, Vol. 38, pp. 183-189, 2000. Matched filtering has been shown to maximize the received signal-to-noise ratio (SNR) in the presence of white Gaussian noise by convolving the transmitted signal with the received radar return signal. One can represent matched filtering in the digital domain as the filtering operation{circumflex over (x)}MF(l)=sH{tilde over (y)}(l),  (1)where {circumflex over (x)}MF(l), for l=0, . . . , L−1, is the estimate of the lth delayed sample of the system impulse response, s=[s1 s2 . . . sN]T is the length-N sampled version of the transmitted waveform, {tilde over (y)}(l)=[y(l) y(l+1) . . . y(l+N−1)]T is a vector of N contiguous samples of the received return signal, and (•)H and (•)T are the conjugate transpose (or Hermitian) and transpose operations, respectively. Each individual sample of the return signal can be expressed asy(l)={tilde over (x)}T(l)s+v(l),  (2)where {tilde over (x)}(l)=[x(l) x(l−1) . . . x(l−N+1)]T consists of samples of the true system impulse response and v(l) is additive noise. The matched filter output can therefore be written as{circumflex over (x)}MF(l)=sHAT(l)s+sHv(l),  (3)where v(l)=[v(l) v(l+1) . . . v(l+N−1)]T and                               A          ⁡                      (            l            )                          =                  [                                                                      x                  ⁡                                      (                    l                    )                                                                                                x                  ⁡                                      (                                          l                      +                      1                                        )                                                                              …                                                              x                  ⁡                                      (                                          l                      +                      N                      -                      1                                        )                                                                                                                        x                  ⁡                                      (                                          l                      -                      1                                        )                                                                                                x                  ⁡                                      (                    l                    )                                                                              ⋰                                            ⋮                                                                    ⋮                                            ⋰                                            ⋰                                                              x                  ⁡                                      (                                          l                      +                      1                                        )                                                                                                                        x                  ⁡                                      (                                          l                      -                      N                      +                      1                                        )                                                                              …                                                              x                  ⁡                                      (                                          l                      -                      1                                        )                                                                                                x                  ⁡                                      (                    l                    )                                                                                ]                                    (        4        )            is a collection of sample-shifted snapshots (in the columns) of the impulse response.
From (4), it is obvious that estimation via matched filtering will suffer from spatial ambiguities (also known as range sidelobes in the radar vernacular) due to the influence from neighboring impulse response coefficients (i.e. the off-diagonal elements of A(l)). To alleviate this effect, Least-Squares (LS) solutions have been proposed, e.g. in M. H. Ackroyd and F. Ghani, “Optimum mismatched filters for sidelobe suppression,” IEEE Trans. Aerospace and Electronic Systems, Vol. AES-9, pp. 214-218, March 1973; T. Felhauer, “Digital Signal Processing for Optimum Wideband Channel Estimation in the Presence of Noise,” IEE Proceedings-F, Vol. 140, No. 3, pp. 179-186, June 1993; S. M. Song, W. M. Kim, D. Park, and Y. Kim, “Estimation theoretic approach for radar pulse compression processing and its optimal codes,” Electronic Letters, Vol. 36, No. 3, pp. 250-252, February 2000; B. Zrnic, A. Zejak, A. Petrovic, and I. Simic, “Range sidelobe suppression for pulse compression radars utilizing modified RLS algorithm,” Proc. IEEE Int. Symp. Spread Spectrum Techniques and Applications, Vol. 3, pp. 1008-1011, September 1998; and T. K. Sarkar and R. D. Brown, “An ultra-low sidelobe pulse compression technique for high performance radar systems,” in Proc. IEEE National Radar Conf., pp. 111-114, May 1997. LS solutions decouple neighboring impulse response coefficients which have been smeared together due to the temporal (and hence spatial) extent of the transmitted waveform. The LS solution models the length-(L+N−1) received return signal asy=Sx+v,  (5)where x=[x(0) x(1) . . . x(L−1)]T are the L true impulse response coefficients that fall within the data window, v=[v(0) v(1) . . . v(L+N−2)]T are additive noise samples, and the convolution of the transmitted waveform with the system impulse response is approximated as the matrix multiplication                               S          ⁢                                           ⁢          x                =                              [                                                                                s                    1                                                                    0                                                  …                                                  0                                                                              ⋮                                                                      s                    1                                                                                                                                                             ⋮                                                                                                  s                    N                                                                    ⋮                                                  ⋰                                                  0                                                                              0                                                                      s                    N                                                                                                                                                                                 s                    1                                                                                                ⋮                                                                                                                                           ⋰                                                  ⋮                                                                              0                                                  …                                                  0                                                                      s                    N                                                                        ]                    ⁢                      x            .                                              (        6        )            The LS model of (6) is employed extensively in radar pulse compression, seismic estimation, e.g. as described in R. Yarlaggadda, J. B. Bednar, and T. L. Watt, “Fast algorithm for lp deconvolution,” IEEE Trans. Acoustics, Speech, and Signal Processing, Vol. ASSP-33, No. 1, pp. 174-182, February 1985; and ultrasonic non-destructive evaluation, e.g. as in M. S. O'Brien, A. N. Sinclair, and S. M. Kramer, “High resolution deconvolution using least-absolute-values minimization,” Proc. Ultrasonics Symposium, pp. 1151-1156, December 1990; and D.-M. Suh, W.-W. Kim, and J.-G. Chung, “Ultrasonic inspection of studs (bolts) using dynamic predictive deconvolution and wave shaping,” IEEE Trans. Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 46, No. 2, pp. 457-463, March 1999. The general form of the LS solution is{circumflex over (x)}LS=(SHS)−1SHy.  (7)
For the received signal model of (5), it can be shown that the LS solution of (7) is optimal in the mean-square error (MSE) sense when the additive noise is white. However, upon further inspection one finds that the LS received signal model does not completely characterize the received return signal because it does not account for the convolution of the transmitted waveform with impulse response coefficients x(l) prior to l=0. The result is that the presence of a significant impulse response coefficient within N−1 samples prior to x(0) can cause severe mis-estimation of the desired coefficients within the data window.
There is, therefore, a need for a predictive deconvolution system with improved robustness, accuracy, and resolution.